Singular limiting induced from continuum solutions and the problem of dynamic cavitation

TL;DR

This paper introduces the concept of slic-solutions to address non-uniqueness and energy considerations in cavitating solutions of elastodynamics, revealing an energetic cost for cavity formation that affects solution stability.

Contribution

The paper proposes slic-solutions as a new framework to incorporate energy costs in cavitation problems, resolving paradoxes related to non-uniqueness and energy in continuum mechanics.

Findings

Cavitation solutions have an associated energetic cost when modeled as slic-solutions.

Accounting for this cost shows cavitating solutions have higher energy than homogeneous states.

The slic-solution concept applies to fracture onset and gas dynamics with vacuum.

Abstract

In the works of K.A. Pericak-Spector and S. Spector \cite{ps88, ps98} a class of self-similar solutions are constructed for the equations of radial isotropic elastodynamics that describe cavitating solutions. Cavitating solutions decrease the total mechanical energy and provide a striking example of non-uniqueness of entropy weak solutions (for polyconvex energies) due to point-singularities at the cavity. To resolve this paradox, we introduce the concept of singular limiting induced from continuum solution (or slic-solution), according to which a discontinuous motion is a slic-solution if its averages form a family of smooth approximate solutions to the problem. It turns out that there is an energetic cost for creating the cavity, which is captured by the notion of slic-solution but neglected by the usual entropic weak solutions. Once this cost is accounted for, the total mechanical energy of the cavitating solution is in fact larger than that of the homogeneously deformed state. We also apply the notion of slic-solutions to a one-dimensional example describing the onset of fracture, and to gas dynamics in Langrangean coordinates with Riemann data inducing vacuum in the wave fan.